How can I make vertices on a cylinder create a circle?
You may use To Sphere operation (as @Dontwalk pointed), but Loop Tool's Circle creates much better result.


Activate the Loop Tools add on in User Preferences (Ctrl+Alt+U). Apply the transformations to the object (Ctrl+A-->Rot&amp;Scale). Select the edges on a cylinder you want to transform to a circle and snap the cursor to it (Shift+S). Press W-->LoopTools-->Circle. Finally uncheck the Flatten checkbox in a Circle panel of a Tool Shelf (T).


You may then extrude the circle inwards as was shown in the tutorial.